Random Sampling

Gathering Data

There are two ways to gather data from a population:

1. Census
2. Sample

A census means to gather data from every member from a population, whilst a sample means to gather data from a group of members from within that population.

Here are the Pros and Cons for each method for the perspective from the business gathering data (this may be questioned):

Central Limit Theorem

When there are more then thirty datapoints in a sample or census, the data increasingly approaches a normal distribution. This theorem is named 'the central limit theorem'. This is very important to know, as it allows us to delve deeper into this distribution, and using our classpad/CAS calculator, extrapolate information from themeasured population.

This theorem has several assumptions:

1. There are atleast 30 datapoints
2. The data must be sampled at random
3. The datapoints must be independant, and not interfer with eachother
4. Sample sizes must be less then or equal to 10% of the population given sampling is done without replacement (you do not need to know this)

We will look more about this later on when it becomes relavent. For now keep it in the back burner.

Bias

In statistics, a bias refers to any unrepresentative samples or sampling/measurement errors that result in an overestimation or underestimation of a certain population parameter's measurement.

In short, a bias is some incorrect sampling methodology that results in your statistics being WRONG!

Sources of Bias:

Selection Bias:

A selection bias is the fault of the sampler (the one who selects members for the study), who may select groups of individuals that all differ systematically from the broader population. For example, selecting only participants who wake up at 12am to complete a study about 'good organisation', or 'healthy lifestyles'. This results in a sample that differs substantially from the population.

Voluntary Bias:

This form of bias arises from the participation of only members with strong views about a particular subject. This results in a more polarised dataset that does not accurately reflect a population with more broad opinions that arn't all 'hot-takes'.

Acceptability Bias:

This one is mad important. You see this concept everwhere. This defines the modern ostraical world, as disgusting, horrific and orwellian as it is. And it truly is. A person may answer a survey or questionaiire, especially when not anonymous, with answers that fit pop culture responses that they percieve would make them more acceptable by others. The goal of increasing ones social desirability may, for many people (most young people nowadays), will be more valuable then applying their own beliefs. As result, the sample does not reflect the population.

Non-Response Bias:

The people that decide to not respond to a survey or study may all have similar viewpoints. This results in a severe decrease in datapoints of like quality. Thus, the sample does not accurately reflect the population.

Agreeability Bias:

This one is optional. Agreeability is a personality trait defined by a persons fear of confrontation, resulting in a person with high agreeability agreeing with most things without giving any thought. Just saying 'no' means opposing something. This bias has it's roots in both the acceptability bias and response bias, and all around is a bias of one's personality traits.

As you can see, there is sooooo many oppurtunities for bias. The truth is, the goal of a sampler is never to negate bias, but to minimise it such that the sample most accurately reflects the population.

Sample Proportions

A sample proportion simply refers to the proportion of a certain metric against the entire sample. Population proportion on the other hand, refers to the proportion of a certain metric against an entire population.

2) Using the above table:

A population proportion can be used to form a binomial distribution, where the conditions are a measure of the sample size less then thirty.

REMEMBER THAT BINOMIAL DISTRIBUTION FORMS ONLY WITH SMALL SAMPLE SIZES

Sample size < 30: Binomial Distribution

For instance, 70.9% of Australian citizens are born in Australia, whilst the other 29.1% were born overseas. Given any sample size, the proportion of Aussies born overseas will be static: p̂ = 0.291.

Using the Binomial CDF eActivity on the Classpad, we can create a table of the sample:

We can form another, more useful table by transforming the number \(x\) into the sample probability using the proportion formula \(p̂ = \frac{x}{5}\):

The distribution of X, which was created using the population proportion (a binomial varaible). Features the same formulas as other binomial distributions:

Using the formula for sample proportions, we can create formulas for the proportion binomial distribution:

Sample size ≥ 30: Normal Distribution and Sample Distribution

As the sample size increases, the graph of the distribution \(y = Bin(X = x)\) approximates normal distribution. We can only be confident that this occurs when:

1. \(n \ge 30\)
2. \(np \ge 15\)
3. \(n(1-p) \ge 15\)

I will be graphing the probability P against x. This was achieved by using the Binomial CDF eActivity and cycling through the Lower and Upper limits (values for x):

As you can see, each graph will have a mean of np

Evidently, as the sample size \(n\) increases, it better and better approximates normal distribution (central limit theorem). It will never reach this, as the binomial distribution will never be continous. However, the shape is well enough to perform calcs with normal distribution when the sample size is above 30.

This is the formula for the sample distribution \(\hat{p}\)

\(\hat{p} \sim N \left(p, \sqrt{\frac{p(1-p)}{n}}^2 \right)\)

3) Pencils are produced by a company in large batches, which are then later refined to meet standards, then sorted into packets. A batch is used to produce a sample of 50 pencils. Out of this sample, 27 pencils have snapped tips.